Angle sum and difference formulas

Let’s see how we can learn it

1.In sin , we have sin cos.

In cos , we have cos cos, sin sin

In tan , we have sum above, and product below

2.For sin (x + y), we have + sign on right..

For sin (x – y), we have – sign on right right.

For cos, it becomes opposite

For cos (x + y), we have – sign on right..

For cos (x – y), we have + sign on right right.

For tan (x + y), numerator is positive & denominator is negative

For tan (x – y), numerator is negative & denominator is positive

Let’s take x = 60°, y = 30° and verify

sin (x + y) = sin x cos y + cos x sin y

sin (60° + 30°) = sin 60° cos 30° + cos 60° sin 30°

sin (90°) = (√3/2) × (√3/2) + (1/2)  × (1/2)

1 = 3/4 + 1/4

1 = 4/4

1 = 1

Hence verified

sin (x – y) = sin x cos y – cos x sin y

sin (60° – 30°) = sin 60° cos 30° – cos 60° sin 30°

sin (30°) = (√3/2) × (√3/2) + (1/2)  × (1/2)

1/2 = 3/4 – 1/4

1/2 = 2/4

1/2 = 1/2

Hence verified

cos (x + y) = cos x cos y – sin x sin y

cos (60° + 30°) = cos 60° cos 30° – sin 60° sin 30°

cos (90°) = (1/2) × (√3/2) – (√3/2) × (1/2)

0 = √3/4 – √3/4

0 = 0

Hence verified

cos (x – y) = cos x cos y + sin x sin y

cos (60° – 30°) = cos 60° cos 30° + sin 60° sin 30°

cos (30°) = (1/2) × (√3/2) +  (√3/2) × (1/2)

√3/2 = √3/4 +  √3/4

√3/2 = 2 × √3/4

√3/2 = √3/2

Hence verified